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Bi-twin Chain
In number theory, a bi-twin chain of length ''k'' + 1 is a sequence of natural numbers : n-1,n+1,2n-1,2n+1, \dots, 2^k n - 1, 2^k n + 1 \, in which every number is prime.Eric W. Weisstein, ''CRC Concise Encyclopedia of Mathematics'', CRC Press, 2010, page 249. The numbers n-1, 2n-1, \dots, 2^kn - 1 form a Cunningham chain of the first kind of length k + 1, while n+1, 2n + 1, \dots, 2^kn + 1 forms a Cunningham chain of the second kind. Each of the pairs 2^in - 1, 2^in+ 1 is a pair of twin primes. Each of the primes 2^in - 1 for 0 \le i \le k - 1 is a Sophie Germain prime and each of the primes 2^in - 1 for 1 \le i \le k is a safe prime. Largest known bi-twin chains ''q''# denotes the primorial 2×3×5×7×...×''q''. , the longest known bi-twin chain is of length 8. Relation with other properties Related chains * Cunningham chain Related properties of primes/pairs of primes * Twin primes * Sophie Germain prime is a prime p such that 2p + 1 is also prime. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eric W
The given name Eric, Erich, Erikk, Erik, Erick, or Eirik is derived from the Old Norse name ''Eiríkr'' (or ''Eríkr'' in Old East Norse due to monophthongization). The first element, ''ei-'' may be derived from the older Proto-Norse ''* aina(z)'', meaning "one, alone, unique", ''as in the form'' ''Æ∆inrikr'' explicitly, but it could also be from ''* aiwa(z)'' "everlasting, eternity", as in the Gothic form ''Euric''. The second element ''- ríkr'' stems either from Proto-Germanic ''* ríks'' "king, ruler" (cf. Gothic ''reiks'') or the therefrom derived ''* ríkijaz'' "kingly, powerful, rich, prince"; from the common Proto-Indo-European root * h₃rḗǵs. The name is thus usually taken to mean "sole ruler, autocrat" or "eternal ruler, ever powerful". ''Eric'' used in the sense of a proper noun meaning "one ruler" may be the origin of ''Eriksgata'', and if so it would have meant "one ruler's journey". The tour was the medieval Swedish king's journey, when newly elected, to s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cunningham Chain
In mathematics, a Cunningham chain is a certain sequence of prime numbers. Cunningham chains are named after mathematician A. J. C. Cunningham. They are also called chains of nearly doubled primes. Definition A Cunningham chain of the first kind of length ''n'' is a sequence of prime numbers (''p''1, ..., ''p''''n'') such that ''p''''i''+1 = 2''p''''i'' + 1 for all 1 ≤ ''i'' < ''n''. (Hence each term of such a chain except the last is a , and each term except the first is a ). It follows that : or, by setting (the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Primes
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sophie Germain Prime
In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. One attempt by Germain to prove Fermat’s Last Theorem was to let ''p'' be a prime number of the form 8''k'' + 7 and to let ''n'' = ''p'' – 1. In this case, x^n + y^n = z^n is unsolvable. Germain’s proof, however, remained unfinished. Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theorem which states that if ''p'' is an odd prime and 2''p'' + 1 is also prime, then ''p'' must divide ''x'', ''y'', or ''z.'' Otherwise, x^n + y^n \neq z^n. This case where ''p'' does not divide ''x'', ''y'', or ''z'' i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Safe Prime
In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. One attempt by Germain to prove Fermat’s Last Theorem was to let ''p'' be a prime number of the form 8''k'' + 7 and to let ''n'' = ''p'' – 1. In this case, x^n + y^n = z^n is unsolvable. Germain’s proof, however, remained unfinished. Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theorem which states that if ''p'' is an odd prime and 2''p'' + 1 is also prime, then ''p'' must divide ''x'', ''y'', or ''z.'' Otherwise, x^n + y^n \neq z^n. This case where ''p'' does not divide ''x'', ''y'', or ''z'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
PrimeGrid
PrimeGrid is a volunteer computing project that searches for very large (up to world-record size) prime numbers whilst also aiming to solve long-standing mathematical conjectures. It uses the Berkeley Open Infrastructure for Network Computing (BOINC) platform. PrimeGrid offers a number of subprojects for prime-number sieving and discovery. Some of these are available through the BOINC client, others through the PRPNet client. Some of the work is manual, i.e. it requires manually starting work units and uploading results. Different subprojects may run on different operating systems, and may have executables for CPUs, GPUs, or both; while running the Lucas–Lehmer–Riesel test, CPUs with Advanced Vector Extensions and Fused Multiply-Add instruction sets will yield the fastest results for non-GPU accelerated workloads. PrimeGrid awards badges to users in recognition of achieving certain defined levels of credit for work done. The badges have no intrinsic value but are valued by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primecoin
Primecoin (Abbreviation: XPM; sign: Ψ) is a cryptocurrency that implements a proof-of-work system that searches for chains of prime numbers. History Primecoin was launched in 2013 by Sunny King, who also founded peercoin. Unlike other cryptocurrencies, which are mined using algorithms that solved mathematical problems with no extrinsic value, mining Primecoin involves producing chains of prime numbers (Cunningham and bi-twin chains). These are useful to scientists and mathematicians and meet the requirements for a proof of work Proof of work (PoW) is a form of Cryptography, cryptographic proof (truth), proof in which one party (the ''prover'') proves to others (the ''verifiers'') that a certain amount of a specific computational effort has been expended. Verifiers can s ... system of being hard to compute but easy to verify and having an adjustable difficulty. Shortly after its launch, some trade journals reported that the rush of over 18,000 new users seeking to mine Pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310. The first five primorials are: : 2, 6, 30, 210, 2310 . The sequence also includes as empty product. Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twin Primes
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term ''twin prime'' is used for a pair of twin primes; an alternative name for this is prime twin or prime pair. Twin primes become increasingly rare as one examines larger ranges, in keeping with the general tendency of gaps between adjacent primes to become larger as the numbers themselves get larger. However, it is unknown whether there are infinitely many twin primes (the so-called twin prime conjecture) or if there is a largest pair. The breakthrough work of Yitang Zhang in 2013, as well as work by James Maynard, Terence Tao and others, has made substantial progress towards proving that there are infinitely many twin primes, but at present this remains unsolved. Properties Usually the pair (2, 3) is not considered to be a pair of twin primes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Safe Prime
In number theory, a prime number ''p'' is a if 2''p'' + 1 is also prime. The number 2''p'' + 1 associated with a Sophie Germain prime is called a . For example, 11 is a Sophie Germain prime and 2 × 11 + 1 = 23 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. One attempt by Germain to prove Fermat’s Last Theorem was to let ''p'' be a prime number of the form 8''k'' + 7 and to let ''n'' = ''p'' – 1. In this case, x^n + y^n = z^n is unsolvable. Germain’s proof, however, remained unfinished. Through her attempts to solve Fermat's Last Theorem, Germain developed a result now known as Germain's Theorem which states that if ''p'' is an odd prime and 2''p'' + 1 is also prime, then ''p'' must divide ''x'', ''y'', or ''z.'' Otherwise, x^n + y^n \neq z^n. This case where ''p'' does not divide ''x'', ''y'', or ''z'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
